This month marks the first anniversary of
Cribbage Forum, which went online in January 2000. In observance of this
milestone, I'm presenting the first installment of Schell's mailbag,
devoted to your questions and comments.

Rules

David Steinbrenner from Wynnewood, Pennsylvania writes:

"Please clear up an ongoing rules 'discussion' I am having with my
father and brother-in-law. We are avid cribbage fans and enjoy the game,
and the fellowship the game brings, immensely. Upon reaching 31 exactly
with the final last card, what is pegged? This may seem like an obvious
question, but reading many sets of rules has not cleared it up. I believe
three points should be pegged: two for 31 and one for last card. My
cribbage mates believe only two should be pegged, not counting the one for
last card. I hope my question is clear. I would appreciate your learned
opinion. Thank you, and I enjoyed your Web site."

David, the rules for scoring a go or last card are probably
the most confusing in cribbage. The answer to your question is: 31 is worth
just two points, even if it comes on the last card played. I'm afraid your
relatives are right!

Not every rule in cribbage is logical, but this one does have a rationale
behind it, which you can grasp if you accept that a go and last
card are basically the same thing. In effect, you're awarded one point
when your opponent is unable to make a legal play, either because:

he has no cards remaining that would keep the count within 31 (a go),
or

he has no cards remaining at all (last card)

If you play a card that makes the count 31, what you get is:

one point for either go or last card, plus

one "bonus" point for hitting 31 exactly

So the two points you peg for 31 already include the point for go
or last card. I hope this explanation helps.

I'm glad you enjoy the game, and that it's a source of fellowship for you
and your associates. This, along with a degree of mental conditioning, is
the most important thing you can get out of cribbage!

Discarding

"I decided after my last game, when my opponent asked me if we were
playing lowball, that I was going to have to learn how to play better
cribbage.

OK, I got 2
4 5
8 10
J.
It's my deal. I threw the 10-J. How would you have
played it?"

First off, let me say that any question about discarding must take
board position into consideration. Different scores often call for
discarding the same hand in different ways (and as we'll soon see,
2-4-5-8-10-J is a good example of this). Without knowing what the
score was when you got this hand, I can only give you general advice.

In most situations, an expert player will toss either 2-4
or 2-8 from this hand. Let's see how these two discards
perform relative to each other, and relative to the other alternatives,
including your own 10-J toss:

Keep:

Toss:

Average
hand:

Chambers
average:

Average crib:

Expected average:

Hessel

Colvert

Ras

Hessel

Colvert

Ras

2-4-5-8

10-J

5.13

6.43

4.64

4.5

4.76

9.77

9.63

9.89

2-4-8-10

5-J

2.35

5.61

7.04

6.9

7.09

9.39

9.25

9.44

2-5-8-10

4-J

6.26

7.09

3.89

3.8

3.98

10.15

10.06

10.24

4-5-10-J

2-8

7.22

7.78

3.58

3.6

3.82

10.80

10.82

11.04

5-8-10-J

2-4

6.57

7.26

4.51

4.4

4.64

11.08

10.97

11.21

The most important number in evaluating discards as dealer is the
expected average, which indicates the average combined value of your
hand and crib after the cut. As you can see, the 2-4 and
2-8 tosses are clearly at the head of the pack, and either
one is a good choice if you're playing on or playing cautious offense.
Tossing 2-4 returns a higher expected average than tossing
2-8, although its lower Chambers average indicates that it
carries more risk, since it places more of its scoring potential in the
crib, which is subject to your opponent's balking efforts. Tossing
2-8 will get you about ¼ point less between hand and crib, but it
leaves you with 4-5-10-J, which is a slightly better
offensive pegging hand due to its two sets of touching cards.
4-5-10-J pegs one point more than 5-8-10-J if pone
has four ten-cards. And you might even get lucky and trap pone's 6
for a 15-6 at the end of the play. If you're playing off however, definitely
keep 5-8-10-J. The 8 gives you a comfortable
reply to a low card lead, something that's hard to defend against holding
4-5-10-J.

Note that if you apply the addition
method to this hand, you'll be correctly guided to the 5-8-10-J
keep, since among the three keeps worth four points in the hand, this is the
one with the highest average crib.

Of course as I mentioned earlier, board position trumps everything. If
the score is 113*-108, for example, your best hand is probably
2-5-8-10, keeping four well-spaced cards for defense (a lone
J is a defensive liability, so I prefer to keep the 10,
even though the J would give me a safe play on an 8
lead). If the score is 118*-118, then keep 2-4-5-8, your best
offensive pegging hand.

For more about discarding as dealer, have a look at the
Discarding to your crib article. If you're
not familiar with terms such as average hands, expected averages and
Chambers averages, I suggest you review the series of Cribbage Forum
articles on calculating discards, beginning with How to analyze discards, part 1.

Speaking of discarding, Life Master George "Ras" Rasmussen sent me this
in October 2000:

"Mike, you have done a great job here in presenting material
relative to discarding. Your format is enlightening and your explanations
where offered seem logical. I like it very much. This is a great service
to those who will take the time to review and apply the concepts to actual
play. Gee, I am off to a great start on the new season with 209 master
rating points after the first month. First time ever that I've won two
Mains in two months. I intend to fully follow your guides to discarding.
Thanks much for sharing."

Hey Ras, thanks for the kind words, and congratulations on your fast
start to the 2000-2001 tournament season! Everyone who follows cribbage
closely knows about your tremendous contributions to cribbage theory, and
your role in popularizing the game here in North America.

Pegging

Several readers wrote in with pegging vignettes. A
Yahoo! Games player
named niner1 writes:

"I would like to say I was impressed with your info on cribbage. I
just started at Yahoo! Games and went 20-1, so I'm a pretty good player
myself. A lot of it was luck, but that one game I did lose was to a
opponent that was very aggressive about pegging on my first card. What
would you lead if you didn't have a pair, say from 2-8-9-J?
I normally would lead a 2 realizing the opponent
might throw down a 9 to try to make me hold my
ten-card and play another low card. But if I pair the 9
I'm in big trouble if dealer has another 9."

Hey, congratulations on your fast start at Yahoo! Regarding your
question: as you've apparently noticed, the problem with leading a 2
from 2-8-9-J is that you're trapped on a 9 or
10 reply. That's why the 8 is a better lead in
most situations. If dealer takes a 15-2 with a 7, you can
either take the run with your 9 (if you're playing on) or
break with your 2 (if you're playing off). On a ten-card
reply, you can either play your J (perhaps pairing dealer's
J) and hope for a go, or else you can break with the
2, keeping the count under 21 and preventing dealer from
getting a three-on-one on the second play series if he has nothing but high
cards.

If you're playing desperation offense or desperation defense, however,
then the 2 is a good lead. Say the score is 114-109*, and
you've cut the right 10. You need to peg two points to win,
and you can assume your opponent will be playing defense. Lead your 2.
It could well catch an 8, 9 or even a 5
in reply (a 3, 4 or J would be
nice too, but unlikely). If you can't score off dealer's first card, try to
save your 8-9 for last, hoping to trap dealer into a run
later. Now imagine the same hand and cut, but at 116-118*. This time you
have enough points in your hand to go out, but dealer only needs to peg
three to win. Since you can't afford to give up even a two-point score, you
should lead your 2, the card dealer is statistically least
likely to be able to score on.

"The score is 116*-116. What would you choose to be dealt if you
know nothing about your opponent?"

Pone has a better than 80% chance of holding a hand worth at least five
points after the cut, so I'm going to try to win by pegging out. The best
hand I can imagine in this situation is A-A-A-8. I'd play the
8 on anything but an A lead, and hope to run
the three As for a triple. If pone is nice enough to cut me a
J, all I'll have to do is run two of the As to
win. Another good hand to hold needing to peg five to win would be something
like 4-5-6-7. This scores two points on anything but an
A, 2 or 3 lead, and has lots of
potential for pairs, 15s, and runs later in the pegging. If I can find just
two two-point scores, then I'll win with last card — provided of course,
that pone doesn't peg out first.

"It was with some interest that I read your article on leading a 5. About
five years ago when I was debugging my pegging algorithm, HALSCRIB led a
5 quite often. So I was not very careful in one game
where I had A-4-J-Q and HAL led a 5.
I played Q and HAL played a 9.
I carelessly played a 4 and then the roof fell in.
HAL pegged 31 for 8:

5Q (15-2) 94AA (30-2) A (31-8)

I was so impressed with this play that I planned to use it in my
next tournament. The very first hand of the first game of a tournament in
Buffalo (at the Air Force base, the year of the Hale-Bopp comet) I
happened to play Jim Bough, a veterinarian from Grand Rapids whose wife
had purchased the DOS version of HALSCRIB years earlier. He didn't fall
for the trap, but we had a good laugh about it after the hand was over. I
still use the play (when playing offense) because most good players like
yourself avoid playing a ten-card. In that same tournament, I held 6-6-7-7 as dealer (playing offense). My opponent led a
J from 5-5-J-J. Needless to
say, I pegged seven points after he played his other J.
(When playing defense, I will lead a 5 from 5-5-x-x — another play I learned from HALSCRIB). I was
the last qualifier and if I had known then what I know now, I may have won
instead of finishing 2nd overall."

Hal, it must be particularly gratifying to learn a trick or two from your
"baby". We'll all look forward to your continued work on HALSCRIB!

Incidentally, on the topic of leading a 5, George
Rasmussen has the following to say in the November 2000 issue of Cribbage
World:

"Have you ever faced the dilemma of holding 5-5-5-x?
And it's your lead? Do you lead a 5 or the ten-card?

I have analyzed this situation, and it is far better to lead a 5!

If you lead your ten-card, the pegging may very well go:

10453 (22-3) 54 (31-5)

Your opponent scores eight points. And you're stuck leading your
last 5 to begin a new sequence. By leading a 5 [to begin with], you will probably be burned by only a
15-2, and a possible second 15-2, risking four points."

Personal database

Duane Stansbie from Toowoomba, Australia asks:

"Reading through your pages you mention keeping your own cribbage
database. What do you record in it? The system I am trying to adopt is to
record cards dealt, and cards discarded from both players. After the game
I'm then trying to see what mistakes were made. You advocate having your
database available during play. Do you have a database which says 'from
this hand discard this, and lead this'?"

Duane,

My database has literally hundreds of records, since I use it to keep
track of published hands analyzed by myself and other authors. Since this
gets a bit unwieldy, I've flagged 100 or so of these records as being
particularly important. These are the entries I review regularly to
reinforce essential ideas. The others I look at only occasionally. I keep my
database in an Excel spreadsheet, organized into pone and dealer
sections, with subsections for four- and six-card hands. Four-card hands are
included mainly to illustrate pegging tactics, while the six-card hands
mainly deal with discarding problems and board strategy.

Like you, I try to analyze my games afterwards, particularly when I play
against a computer program like
Cribbage
for Windows 97, which has a logging feature. If I find an
improvement in my play, or a succinct demonstration of an important concept,
I'll add a new record to my database.

Here are a couple sample entries from my database. The first is filed
under pone six-card hands:

A-A-A-4-4-4

If you were dealt this hand, would you keep A-A-A-4 or
A-4-4-4? Both are worth six points going in, and both will
fetch twelve points on a ten-card cut. A-A-A-4looks
best since it has the lowest pip count, and generally lower cards are
better. But it turns out A-4-4-4 is superior, since it gets
twelve points on a 6, 7 or ten-card cut, while
A-A-A-4 only gets twelve on a 9 or ten-card
cut. A-A-4-4 is only worth considering for defense (A-4
is a tad safer toss to opponent's crib than A-A or 4-4).

Here's another entry, from the dealer four-card hands section:

3-3-4-J

I added this record as a result of a game I played at my Grass Roots
club. The pegging started:

6J10 ?

I played my 4 here to bring the count closer to 31. But
this left me holding an inflexible 3-3. A better idea would
have been to play one of the 3s instead. This slightly
increases the chances of giving up a go (it's vulnerable to an
A or 2, instead of just an A), but it
keeps the 5 trap alive. As it turned out, my opponent's last
two cards were 5-6. If I'd had been holding 3-4
instead of 3-3, I'd have gotten a four-card run. Moral: as
dealer it's worth taking small risks to save a 5 trap for the
end of the play. Since pone will be naturally reluctant to toss you a
5, he'll often be holding one even if it doesn't match his other
cards. As a result, a trap laid for a 5 has an unusually good
chance of success.

Now I don't recommend you keep hundreds of records as I do. And it's not
necessary to organize your database the same way I do. You could sort your
entries by hand type, grouping together mid-card hands, or hands with
5s and ten-cards, or split lo-hi hands such as A-4-x-x
or 2-2-3-x. You can also organize your database by theme.
For example, a section on 5 traps could include hands like
2-3-4-9, 4-5-6-6 and 6-7-8-x,
in addition to the above 3-3-4-J example. And you could have
other sections on magic elevens, different kinds of endgames, and so on. I
do suggest, though, that you start with hands and plays that you find
troublesome. As you continue your cribbage growth, add entries that
communicate essential ideas in ways that are meaningful to you. The
A-A-A-4-4-4 example I gave above helps me to remember that lower
cards are not necessarily better, while the 3-3-4-J example
reminds me to stay alert for 5 traps. You can rearrange your
entries as your study focus changes, and delete records that become
redundant or less useful. Soon you'll have developed your own personal
theory of cribbage.

29 hands

Jack Remlin of South Hadley, Massachusetts asks:

"Can you tell me where to find accurate odds of getting a 29 point
hand in cribbage? Are the odds the same for two-player and three-player
games?"

Jack, the mathematical probability of getting a 29 hand in two-player
cribbage has long been known to be 1 in 216,580. But I've never seen an
accurate quoted figure for a 29 hand in three-player cribbage, so I decided
to calculate it for myself. To understand the applicable math, I'll begin by
demonstrating how the 1 in 216,580 figure for two-player cribbage is
derived.

The branch of mathematics known as combinatorics provides a
formula for calculating the number of possible combinations of k
items selected from n total items:

In two-player cribbage, a hand contains six cards selected from 52 total
cards, so according to this formula the total number of possible hands is:

Any six-card hand with the potential for 29 points must include a
specific four-card combination: three 5s and the J
of the missing suit. There are four such combinations available, (one for
each J suit). The fifth and sixth cards can be anything else
except the missing 5 (which must be the starter), so
the number of eligible six-card hands is:

Now 20,358,520 divided by 4,324 is about 4,708.26, so your odds of
drawing an eligible six-card hand are 1 in 4,708.26. Assuming you then keep
5-5-5-J, the odds of cutting the right 5 to it
are 1 in 46. So your odds of getting a 29 hand in two-player cribbage are 1
in:

Or 1 in 216,580.

In three-player cribbage, you're dealt only five cards, so the total
number of possible hands is:

Again, a potential 29 hand must contain one of four possible combinations
of three 5s and the J of the missing suit. The
fifth card can be anything else except the missing 5, so the
number of eligible five-card hands is:

2,598,960 divided by 188 is 13,824.26, so your odds of drawing an
eligible five-card hand are 1 in 13,824.26. Assuming you then keep
5-5-5-J, your chances of cutting the right 5 to it
are 1 in 47 (not 1 in 46 as in the six-card game). So your odds of getting a
29 hand in three-player cribbage are 1 in:

Or 1 in 649,740. The odds are identical in four-player cribbage.

My thanks to Hal Mueller and Nick Wedd for their mathematical guidance.

Oops!

Bill Borghard wrote in with this:

"Terrific Web page! Awesome! I had started doing something similar
to what Hessel had already done [i.e., calculate discard averages]. Kind
of relieved, it would have been a lot of work. I did however, come up with
a slightly different number of possible six-cards hands, ignoring suits. I
have 18,395 instead of the 18,564 number. You can have:

4,1,1 (four of one rank, one of another
and one of yet another)

=

=

858

4,2

=

=

156

3,1,1,1

=

=

2,860

3,2,1

=

=

1,716

3,3

=

=

78

2,1,1,1,1

=

=

6,435

2,2,1,1

=

=

4,290

2,2,2

=

=

286

1,1,1,1,1,1

=

=

1,716

Total:

18,395

Not much difference, but thought I would let you know."

The 18,564 figure was cited in the original posting of my article
Discarding to your crib. You're right: the
number of six-card hands irrespective of suit is actually 18,395. The error
was mine, not Hessel's, and I've updated the article with the correct
figure. Thanks for pointing this out!

Nice words

Linda Falkenstein writes:

"Thanks so much for your pages. It's so well done, exactly the kind
of information I have been looking for. I look forward to your next
update."

And Wilf Kelly, an occasional opponent of mine at
PlaySite writes from
Ontario, Canada:

"I enjoyed reading your articles on your site. I find it very
interesting. I wish we had some tournaments in Canada, like you do in the
States. Keep up the good work, and good luck next time we meet in crib."

Thanks for the kind words Linda and Wilf. As for sanctioned tournaments
in Canada, that's starting to happen, at least out West. The
ACC sanctions two
yearly tournaments in Edmonton, and there was one last year in Prince George
as well. There are also Grass Roots clubs in Alberta and British Columbia.
Hopefully interest will spread eastwards. Perhaps it's time for the ACC to
change its name to better reflect its international scope.

Mike Fetchel from East Hartford, Connecticut writes:

"Just wanted to drop you a line telling you that I really enjoyed
your cribbage site. I've been playing cribbage for over 25 years and just
recently joined the ACC and a Grass Roots club here in East Hartford. I'm
always trying to improve my game and your site has been a big help. Thanks
again and happy pegging."

Mike, I hope you enjoy playing competitive cribbage in the ACC. They're a
tough bunch of players — the world's best. Good luck, and perhaps we'll meet
at the Tournament of Champions in Reno one day!

John T. Irving, an old Internet adversary of mine from Vancouver, British
Columbia, writes:

"Just finished reading your article A course
of study and found it very informative. I plan on reading the
books you suggest and look forward to your future articles on cribbage.
Hope this will help get me to the next level. Look forward to playing
again sometime."

John, you're the only person who ever beat me 6-0 in a Burbs match by
skunking me twice in two games! I see you've pushed your
Zone cribbage rating over
2600, so it looks like your game's stronger than ever. I'm glad the articles
have helped your play, and look forward to our next match.

That's all for now.
Keep in touch
everyone, and we'll do another one of these soon.